Sharp invertibility of random Bernoulli matrices

Let $p\in (0,1\mathrm{/}2)$ be fixed, and let $B_n(p)$ be an $n\times n$ random matrix with i.i.d. Bernoulli random variables with mean $p$. We show that for all $t\ge 0$, [$\mathbb{P}$[s_n(B_n(p)) \le tn^{-1/2}] \le C_p t + 2n(1-p)^{n}

• C_p (1-p-\epsilon_p)^{n},] where $s_n(B_n(p))$ denotes the least singular value of $B_n(p)$ and $C_p,{ϵ}_p>0$ are constants depending only on $p$. In particular, $$\mathbb{P}[B_n(p)\text{ is singular}]=2n(1-p{)}^n+C_p(1-p-{ϵ}_p{)}^n,$$ which confirms a conjecture of Litvak and Tikhomirov. We also confirm a conjecture of Nguyen by showing that if $Q_n$ is an $n\times n$ random matrix with independent rows that are uniformly distributed on the central slice of $\{0,1{\}}^n$, then $$\mathbb{P}[Q_n\text{ is singular}]=(1\mathrm{/}2+o_n(1){)}^n\mathrm{.}$$ This provides, for the first time, a sharp determination of the logarithm of the probability of singularity in any natural model of random discrete matrices with dependent entries.